Create Table goodness of approx for t-test.

Here we create a copy of the table 1.1 of Asymptotic Statistics, A. W. van der Vaart.

Look at the distributions we are working with.

sample.1 <- rnorm(1000, mean = 2, sd = 3)
hist(sample.1)

sample.2 <- rexp(1000, rate = 3)
hist(sample.2)

Generate a collection of data from samples taken.

many.sample.stats <- function(samplefn, samplesize, mu, iterations=1000) {
  dat <- Reduce(rbind, lapply(seq(1, iterations),
                                function(x) {s <- samplefn(samplesize)
                                             data.frame(mean=mean(s),
                                                  sd= sd(s))}))
  vals <- sqrt(samplesize) * (dat$mean - mu) / dat$sd
  vals
}

vals.n <- many.sample.stats(function(ss) {rnorm(ss)}, 1000, 0, 10000)
vals.e <- many.sample.stats(function(ss) {rexp(ss)}, 1000, 1, 10000)

Compute the relevant fraction.

perc.above <- function(bd, ...) {
  vals <- many.sample.stats(...)
  ct <- length(vals)
  ab <- sum(abs(vals) > bd)
  ab/ct
}

Get the data

sizes <- c(5,10,15,20,25,50,100)
bd.n.exact <- sapply(sizes,
                     function(size) 2*(1-pt(1.96, size-1)))
bd.n <- sapply(sizes, 
               function(size)
                 perc.above(1.96,
                            function(ss) {rnorm(ss)},
                            size,
                            0,
                            10000))
bd.e <- sapply(sizes,
               function(size)
                 perc.above(1.96,
                            function(ss) {rexp(ss)},
                            size,
                            1,
                            10000))

Show the table

library(xtable)
print(xtable(data.frame(bd.n.exact, bd.n, bd.e)), type="html")

	bd.n.exact	bd.n	bd.e
1	0.12	0.12	0.18
2	0.08	0.08	0.13
3	0.07	0.07	0.11
4	0.06	0.07	0.09
5	0.06	0.06	0.09
6	0.06	0.06	0.07
7	0.05	0.06	0.06